Scenario: Imagine you are an engineer tasked with designing the logic for a smart home security system. The system involves various sensors and actuators to monitor and control different aspects of the home's security. Your goal is to use Boolean algebra and logic gates to create a logical framework that efficiently represents the conditions and actions for the smart home security system.

a$.$ Apply De Morgan's Theorems to express the negation of the following conditions:

i$)$ Smoke is detected in the kitchen.

ii$)$ Windows are closed.

b$.$ Show the relationship between De Morgan's Theorems and other Boolean algebra laws in your explanations